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Difference between revisions of "Bolzano-Weierstrass theorem"

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Each bounded sequence of numbers contains a convergent subsequence. The theorem applies both to real and complex numbers. It can be generalized to include more-general objects, e.g. any bounded infinite set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016880/b0168801.png" />-dimensional Euclidean space has at least one limit point in that space. There exist analogues of this theorem for even more-general spaces.
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Each bounded sequence of numbers contains a convergent subsequence. The theorem applies both to real and complex numbers. It can be generalized to include more-general objects, e.g. any bounded infinite set in $n$-dimensional Euclidean space has at least one limit point in that space. There exist analogues of this theorem for even more-general spaces.
  
 
The theorem was demonstrated by B. Bolzano [[#References|[1]]]; it was later also independently deduced by K. Weierstrass.
 
The theorem was demonstrated by B. Bolzano [[#References|[1]]]; it was later also independently deduced by K. Weierstrass.

Revision as of 21:14, 20 October 2012

Each bounded sequence of numbers contains a convergent subsequence. The theorem applies both to real and complex numbers. It can be generalized to include more-general objects, e.g. any bounded infinite set in $n$-dimensional Euclidean space has at least one limit point in that space. There exist analogues of this theorem for even more-general spaces.

The theorem was demonstrated by B. Bolzano [1]; it was later also independently deduced by K. Weierstrass.

References

[1] B. Bolzano, Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v. (1817)
How to Cite This Entry:
Bolzano-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bolzano-Weierstrass_theorem&oldid=28588
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article